Mechanical vibrations worksheet with answers

Free printable Mechanical vibrations worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy

A 20 kg mass hangs from a spring with a stiffness of 10 kN/m. Ignore damping. Find the natural circular frequency ω_n in rad/s.

m = 20 kgk = 10 kN/m

ω_n = rad/s

2.Easy

A 20 kg mass hangs from a spring with a stiffness of 50 kN/m. Ignore damping. Find the natural frequency f_n in Hz.

m = 20 kgk = 50 kN/m

f_n = Hz

3.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 50 mm is followed 2 cycles later by a peak of 26.7 mm. Find the damping ratio ζ.

x₀ = 50 mmx₂ = 26.7 mmn = 2

ζ =

4.Medium

In a free-vibration test of a damped mass–spring system, a displacement peak of 25 mm is followed one cycle later by a peak of 15.1 mm. Find the damping ratio ζ.

x₀ = 25 mmx₁ = 15.1 mmn = 1

ζ =

5.Hard

A 10 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 15 kN/m and k₂ = 20 kN/m. The mass stays level. Ignore damping and the masses of the springs. Find the natural frequency f_n of vertical vibration in Hz.

m = 10 kgk₁ = 15 kN/mk₂ = 20 kN/m

f_n = Hz

6.Hard

A 100 kg mass hangs from two springs side by side (in parallel), with stiffnesses k₁ = 10 kN/m and k₂ = 15 kN/m. The mass stays level. Ignore damping and the masses of the springs. Find the natural frequency f_n of vertical vibration in Hz.

m = 100 kgk₁ = 10 kN/mk₂ = 15 kN/m

f_n = Hz

Answer key

1. ω_n = 22.36 rad/s
2. f_n = 7.958 Hz
3. ζ = 0.04986
4. ζ = 0.07999
5. f_n = 9.416 Hz
6. f_n = 2.516 Hz

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